Definition:Counterexample

Definition

Let $X$ be the universal statement:

$\forall x \in S: \map P x$

That is:

For all the elements $x$ of a given set $S$, the property $P$ holds.

Such a statement may or may not be true.

Let $Y$ be the existential statement:

$\exists y \in S: \neg \map P y$

That is:

There exists at least one element $y$ of the set $S$ such that the property $P$ does not hold.

It follows immediately by De Morgan's laws that if $Y$ is true, then $X$ must be false.

Such a statement $Y$ is referred to as a counterexample to $X$.

Examples

Sum of Cubes of Digits

Let $P$ be the statement:

There exists no integer which is the sum of the cubes of its digits.

A counterexample to $P$ is the number $153$, as can be seen in Pluperfect Digital Invariants: $3$ Digits:

 $\ds 153$ $=$ $\ds 1 + 125 + 27$ $\ds$ $=$ $\ds 1^3 + 5^3 + 3^3$

Sum of Sine and Cosine equals $1$

Let $P$ be the statement:

$\forall x \in \R: \cos x + \sin x = 1$

A counterexample to $P$ is the real number $\dfrac \pi 4 \in \R$:

 $\ds \cos \dfrac \pi 4$ $=$ $\ds \dfrac {\sqrt 2} 2$ Cosine of $\dfrac \pi 4$ $\ds \sin \dfrac \pi 4$ $=$ $\ds \dfrac {\sqrt 2} 2$ Sine of $\dfrac \pi 4$ $\ds \leadsto \ \$ $\ds \cos \dfrac \pi 4 + \sin \dfrac \pi 4$ $=$ $\ds \sqrt 2$ $\ds \ne 1$

Also see

• Results about counterexamples can be found here.

Internationalization

Counterexample is translated:

 In German: Gegenbeispiel

Sources

... Any example which in some respect stands opposite to the reals is truly a Gegenbeispiel.